On Approximation for Fractional Stochastic Partial Differential Equations on the Sphere
It provides theoretical convergence guarantees for approximating fractional SPDEs on the sphere, relevant for geophysics and cosmology applications.
The paper derives an exact solution via Karhunen-Loève expansion for a fractional SPDE on the sphere and provides a numerical approximation with proven convergence rates in degree and time, demonstrated on CMB simulations.
This paper gives the exact solution in terms of the Karhunen-Loève expansion to a fractional stochastic partial differential equation on the unit sphere $\mathbb{S}^{2}\subset \mathbb{R}^{3}$ with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen-Loève expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background (CMB) are given to illustrate the theoretical results.